How to find concavity from the first derivative graph?
Understand the Problem
The question is asking how to determine the concavity of a function based on its first derivative graph. To address this, we need to examine the behavior of the first derivative: when it is increasing or decreasing, which indicates where the original function is concave up or concave down.
Answer
Concavity is determined by analyzing intervals of increasing or decreasing behavior in the first derivative graph.
Answer for screen readers
The intervals of concavity depend on the analysis of the first derivative graph.
For example:
- If the first derivative is increasing on the interval $(a, b)$, then the original function is concave up on $(a, b)$.
- If the first derivative is decreasing on the interval $(c, d)$, then the original function is concave down on $(c, d)$.
Steps to Solve
- Analyze the first derivative graph
Look at the first derivative graph of the function. Determine the intervals where the first derivative is increasing or decreasing.
- Identify critical points
Find the critical points where the first derivative changes from increasing to decreasing or from decreasing to increasing. These points are usually where the first derivative is equal to zero or undefined.
- Determine concavity from the first derivative
- If the first derivative is increasing in an interval, then the original function is concave up in that interval.
- If the first derivative is decreasing in an interval, then the original function is concave down in that interval.
- Summarize results
List out the intervals for concavity based on the behavior of the first derivative graph, indicating where the function is concave up and where it is concave down.
The intervals of concavity depend on the analysis of the first derivative graph.
For example:
- If the first derivative is increasing on the interval $(a, b)$, then the original function is concave up on $(a, b)$.
- If the first derivative is decreasing on the interval $(c, d)$, then the original function is concave down on $(c, d)$.
More Information
The concavity of a function can provide insights into its behavior, such as whether it has a local maximum or minimum. When the first derivative is increasing, the slope of the function is getting steeper, indicating concavity upwards. When it is decreasing, the slope is getting shallower, indicating concavity downwards.
Tips
- Confusing the increasing/decreasing nature of the first derivative with that of the original function.
- Not correctly identifying critical points where the behavior of the first derivative changes.
